Photoelastic strain gauges



Aug- 27, 1963 s. REDNER l 3,101,609

PHoToELAsTIc STRAIN GAuGEs l Filed Oct. 16. 1961 q: INVENTOR. I SolomonRedner BY @imam ATT ORNE Y United States Patent C mma Filed Oct. 16,wel, Ser. No. 145,086

3 Claims. (Ci. 7S-88) This invention pertains to improvements inphotoelastic strain gauges which are omnidirectional in their resolutionof workpiece surface strains and to methods for the manufacture of suchgauges.

The Ibasic omnidirectional strain gauge, disclosed and claimed in theleopending application `of Georges Golubovic, Serial No. 701,592, iiledDecember 9, 1957, now Patent 3,034,341, comprises a photoelastic(forcedabirefringent) material testpiece conformed as a dat, c1rcularwasher with a concentric Ecircular aperture. When such a testpiece isattached about its periphery to a workpiece surface and observed bymeans of a simple polarisoope, it produces photoelastic patterns winchresolve the magnitudes and :directions of principal strains at theworkpiece surface. The Golu'bovi-c gauge is a most signiicant advanceover the prior photoelastic strain gauges which respond to straindifferences, `or to a unidirectional strain when only that strain iseffective, without resolution of the variously directed strains in ageneral lbiaxial strain iield. However, the photoelastic patternsproduced by the Golubovic `gauge are somewhat complicated and theirinterpretation requires training and experience.

Therefore, it is a general object of this invention to provide animproved photoelastic ystrain gauge yielding simply and easilyinterpreted photoelastic patterns for the resolution of workpiecesurface strains.

A more specific object is to provide an improved photoelastic straingauge :for the resolution =of workpiece surface strains by means ofphotoelastic patterns which are regular polar-coordinate plots of theprincipal surface strains at the surface of the workpiece, thephotoelastic patterns comprising workpiece surface strain relatedriorced-'birefringence superimposed upon a regular pattern ofibiasabirefringence preformed in the photoelastic testpiece of thegauge.

A further specific object of this invention is to provide an eicient,precise, and inexpensive method for producing a regular pattern ofbias-birefringence in forced-birefringent material testpieces for use inomnidirectional photoelastic strain gauges.

An illustrated embodiment of the improved strain gauges of thisinvention comprises a stratum of forcedbirefringent material shaped nodefine a liat circular testpiece with a concentric aperture and meanscontiguous with the testpiece periphery rigidly attaching the testpieceto the surface of the workpiece, the testpiece having a radiallysymmetrical pattern of bias-birefringence preformed therein, wherebyphotoelastic patterns pro'- duced by the gauge are polar-coordinateplots of the principal strains at the workpiece surface.

According to this invention, a preferred method for producing aperturedtestpieces having radially symmetrical patterns of bias-lbirefringencecomprises the steps of forming, from sheet plrotoelastic material, flatcircular testpiece blanks having concentric apertures, thereafterheating the `blanks to a given temperaturel to above the photoelasticcritical temperature for the material, subjecting the testpiece `blankswhile at the given temperatures to an outwardly radially directedpressure applied at the periphery of the apertures, and subsequentlycooling the blanks below the critical temperature ywhile maintaining theapplication .of the radially directed pressure, whereby substantiallyconcentric patterns of Ibias-'birc- CCl frigence are exhibited by thetestpiece blanks in the absence Iof externally imposed strains.

While the articles and methods of this invention are particularlydefined in the appended claims, a better understanding thereof will `Ibehad upon consideration of the following specification taken inconjunction with the accompanying drawings wherein:

FiG. l and FIG. 2 are plan and cross-section views of a strain gaugeconstructed according to this invention and attached to. a lworkpiecesurface;

FIG. 3 is a graph of a pattern of bias-birefringence I imposed upon thetestpiece of the gauge of FIGS. 1 and FIG. 4 illustrates stress andstrain relationships useful in explanation of the operation of the gaugeof FIGS. 1 and 2;

FIG. 5l and FIG. 6 are plan and cross-section views of preferredpolariscope in place upon the `gauge of FIGS. l and 2;

FIG. 7 illustrates photoelastic fringe pattern variations exhibited bythe :gauges of this invention; and

FIG. 8 depicts an example of apparatus for providing patterns ofbias-birefringence in strain gauge testpiece blanks according to thisinvention.

With reference to FIGS. 1 and 2, a photoelastic strain gauge i0constructed according to this invention comprises a flat washerwshapedtestpiece l2. having an inner radius a, the radius of aperture 14, andan outer radius b. Such a testpiece may be :described as having twonon-intersecting edges-outer periphery 16, aperture per1phery 18.Preferably, testpiece 12 is blanked from forced-birefringent sheetmaterial (Bakelite, or the like) of constant thickness t and havingpolished surfaces which, undisturbed, become the lateral testpiecesurfaces 20 and 22.. As does the Golfulbovic gauge, the omnidirectionalstrain gauge of this invention includes an outer peripheral attachmentmeans 24, an annular layer of cement for example, contiguous with outerperiphery 16 for rigid attachment of testpiece 12 to surface 26 ofworkpiece 28. According to this invention, however, testpiece 12contains a preformed radially symmetrical pattern of bias-birefringencewhich, in the absence of irnposed strains, causes the testpiece togenerate circular isochromatic interference fringes in normal incidencepoe larized light. This bias pattern preferably includes at least oneboundary @fringe position 30 midway between edges 16 and y18.

A photoelastic pattern of isochroma-tic fringes is directly related todifferences between the ordinary and extraordinary indices of refractionof an optically aniso- Tropic material. When refractive indexdifferences are dependent upon externally imposed stresses (or strains)this anisotropy is referred to as force'd-birefringence. When suchdifferences exis-t in a photoelastic material in the absence of externalstimuli, the anisotropy is referred to as bias-birefringence.Photoelastic patterns produced by a given photoelastic test-piece are adirect func-tion of total birefringence, the summation offorced-birefringence and bias-birefringence. As total birefringenceincreases, the color of an associated interference fringe repeatedlyvaries through the spectral wave lengths. A boundry fringe is thewell-defined color bland between the bluegreens of one order and thereds of the next order. Fringe production is well understood in the artof photoelastioity land 4a detailed discussion of the subject may befound in Photoelasticity, by M. M. Frocht, John Wiley and Sons, NewYork, 1941.

A desired pattern of bias-birefringence may be described by theequivalent stress-difference pattern which, when imposed upon a givenforced-birefringent material, would yield a forced-birefringence patternduplicating 3 that pattern fof bias-birefringence. FIG. 3 is a graph lofthe circular-ly symmetrical principal stress difference variation withradial position which is descriptive of the preferred testpiecebias-birefringence. Principal stress difference As varies inversely asthe square `of the distance r from the axis of symmetry of thetestpiece. At radial distance f, As has the rvalue Bf Iwith respect tothe testpiece thickness t which yields the boundary fringe locus 3G)depicted in FIG. 1. Preferably, the As curve varies substantiallylinearly with small radial displacements from f as indicated by theclose approximation -fof the As curve by its tangent in the vicinity ofthat radius. Superimposed forced-bircfringence at a testpiece region (r,see FIG. 1, would translate the s curve vertically (positively ornegatively) tand :the intersection of the resultant curve with the Bf-abscissa would shift radially, resulting in a concomitant radialdisplacement of point f and of curve .30. The radial *displacement ofcurve Sil from its original circle to an ellipse 30', for example, maybe considered :a polar-coordinate plot of forced-birefringence and ofthe external condition responsible for the forcedbirefringence.

FlG. 4 reproduces testpiece i2 bonded at its periphery 16 to workpiecesurface 26 for illustration of the relationship betweenforced-birefringence and workpiece surface strain conditions, whichrelationship is common to the basic Golubovic gauge and to 'the gaugesof this invention. Consider the region (r, 0) at an angular position 0and radial position r. `Outer periphery 16 is strained with theworkpiece surface 26 and aperture periphery i4 is a free boundary. Sincethere can be no substantial radial stress through region (r, 0)perpendicular to the free boundary i4, the -tangential strip 32 `ofincremental width containing the region (r, 0) acts as a simplephotoelastic extensometer parallel with direction :9-90. Theextensometer maximum unit strain e will be equal to the parallelworkpiece strain ed, and fthe perpendicular extensome-ter unit strain e'is the Poisson strain related to e by Poissons ratio np for thetestpiece material: e=-ppe. Forced-birefringence Bm at the region (r,gb) is given by:

where lc is la constant for a given testpiece. Equation I is independentof ea and, therefore, the basic :gauge may be ldescribed as resolvinguniaxial strains in a biaxial strain iield.

A complication of the basic gauge is diagramed at the right in FIG. 4.Along the ydiameter through region (r, 0) the testpiece stress s, in thetangential direction qb varies with radial position according to curve34- and there does exist a small radial stress se varying with radialposition according to curve 36. Curves 34 and 3rd-taken from thetheoretical ydescription of an apertured plate in tension, have been`found empirically to be qualitatively applicable to the boundaryconditions prescribed for testpiece l2. While s., may be neglected atsmall radial distances from the free boundary, the variations contributeto :the kaleidoscope multi-lobed photoelastic patterns of the Golubovicgauge. According to this invention, however, bias-birefringencedescribed in connection with FlG. 3 cooperates rwith imposedforced-birefringence to produce boundry fringe displacements which aredeterminative of principal strain magnitudes and directions.

`For further explanation of the effect of the bias-bircfringence uponthe omnidirectional strain gauges, particular reference is directed toFIGS. 5 and 7. Photoelastic patterns are visualized by means tof apolariscope system, here comprising polariscope 3S, reflector 40, tand alight source, ordinary room illumination for example. Polariscope 38 maycomprise a laminate of plane polarizing sheet material 4Z and aquarter-wave retardation plate 44, attached to a rubber disc 46 whichtits within aperture 14 of testpicce 12. Reflector tft-0 may be arellective coating applied yto the workpiece side `of testpiece 12 orits function may be provided for by a reflecting workpiece surface. Apaper label 48 may be utilized advantageously to provide angularindicia.

FIG. 5 is a plan view tof the )assembled polariscope and gaugeillustrating, by shading 50, a boundary fringe produced in normalincidence light by the bias-birefringence pattern of FIG. 3. Radialindicia are shown as circles `52 scribed or printed on a surface ofpolariscope 44. As workpiece 2S is loaded and strains are developed atworkpiece surface 26, the boundary fringe will shif-t concomitantly withthose strains lfrom its original position 5G` and will assume a quarticform illustrated by `an ellipse at Sli' in FIG. 7.

As explained in connection with FIG. 3, forced-birefringence added tothe preformed bias-birefringence causes radial changes in the locus ofthe boundary fringe. However, as explained in connection with FIG. 4,the forcedbirefringence `at an angular position 6 is proportional to thetangential workpiece `Strain yalong the direction =090. Therefore, achange in the boundary fringe radius along the `direction 0 is a measureof the |workpiece surface strain `along the direction qb at rightyangles to the direction 0.

FIG. 7 illustrates the boundary fringe displacement produced by taworkpiece surface 'strain field in Iwhich the maximum stnain emax is atensile strain `along the `120 direction and the minimum surface strainemu, is a compressive strain along the 30 direction. The directions ofthe principal surface strains are given by mere inspection. The maximumsurface strain is parallel `with the minor axis of the boundary fringeellipse Sil and the minimum principal surface strain is parallel lwiththe major axis of the boundary fringe ellipse 50'.

Evaluation of a principal or intermediate strain vector magnitude isgiven by measurement of the boundary fringe shift along the radiusperpendicular to that vector. The boundary fringe position 50 is,therefore, a polarvcoordinate plot, rotated by of the strain vector 'toFIG. 3, when there is la suiiicient forced-birefringence added to thebias-birefringence to shift the total bircfringence curve verticallyuntil it intersects an abscissa Bf which represents the tota-lbirefringence necessary for production of the boundary fringe `of thenext higher order than that of the residual boundary fringe. Strainmagnitilde values `for the boundary fringe 54 are simply those read fromradial indicia 52 plus the Value of the difference in the radialseparation of the second' fringe from the radial position of the -iirstfringe. This difference must, of course, be recorded before the iirstboundary fringe moves out of the field of View.

The icopending application of F. Zandman and S. Reider, Ser. No.799,798, liled March 16, 1959, and assigned to the same assignee as isthis application, discloses and claims the production of certainspecialized residual or biasing patterns of birefringence and may bereferred to for background information. However, the mechanism ofresidual bias-birefringence patterns may be explained generallyaccording to a diphase theory of the molecular structure offorced-birefringent plastic materials. These materials owe theircharacteristics to two sets of molecular bonds. Primary infusible bondscreate as la first phase a random structural framework throughout thematerial and relatively weak secondary bonds constitute a second fusibleIphase entwined about the iirst phase network. The second phase fuses orbecomes plastic at a moderate temperature, a so-called photoelasticcritical temperature. However, the iirst phase is substantiallyinfusible below the decomposition temperature of the material.

At temperature above the photoelastic critical temperature, the diphaseis a composite of fused material encompassing a structural framework ofunfused material so that external loading forces selectively induceelastic strains `and restoring stresses in the infusible phase. When thetemperature is reduced below the photoelastic critical temperature, thefusible phase congeals yand thereafter opposes relief of the stressesinduced in the infusible phase. Upon subsequent removal of the loadingforces an equilibrium condition is established in which residualstresses persist. The residual stress patterns, and hence the residualbirefringence patterns, is 'geometrically similar to the original stresslpatterns induced :by the loading forces.

Preferred materials for the testpieces of this invention includetransparent polymerized plastic materials of which Bakelite, layglycerin phthallic anhydride, is yan excellent example. Otheradvantageous materials include resins of the type in which alkyds `arecopolymerized with styrene. The photoelastic critical temperature for anapplicable material may be readily determined empirically uponcollection of stress-strain data at a series of temperature conditions.For example, the photoelastic critical temperature of Bakelite isbetween the approximate limits of 230 and 260 F.

For the purposes of this explanation, ra photoelastic criticaltemperature range is `defined as including those temperatures at whichya photoelastic plastic kacts as a diphase and exhibits a substantiallylinear stress-strain relationship under load.

Satisfactory methods for the production :of regular biasbirefringencepatterns in two-edge testpieces, testpieces having two closed randnon-intersecting edges, have not been available prior to this invention.The problem has now been solved by means of the simple, inexpensive, andprecise means and methods to be described in connection with theapparatus illustrated in FIG. 8.

FIG. 8 is a cross-section elevation showing la constant temperaturefurnace 80 whose interior may be cycled precisely to temperatures withinthe photoelastic critical temperature range of a givenforced-birefringent material. 'Iestpieces Iare blanked from auniform-thickness sheet of forced-birefringent material, preferably acast sheet, in order to obviate polishing of their lateral surfaces. Thetestpiece blanks I82 are stacked with interleaved separators 83 on asupport assembly 84 which comprises 4a base 86, a rigid post 8S, and sathin-'walled expandable tube 90, of a rubber-like material. Annularpressure seals 94 and 96 are provided between post 88 and tube 90 bycementing or other means. Communication with the internal void betweenpost 88 and tube 90 is by way of ports 98 in post 88, `and base port1001and 102, the latter being connected with a pressure tube 104 leadingthrough the wall of furnace y80 to a source, not shown, of controlledair pressure. The pressure po applied to the outer peripheries oftestpiece blanks 82 may be atmospheric pressure. The lgenerated pressureis adjusted to net a radially ldirected pressure pi against the innerperipheries of testpiece blanks 82.

Under these conditions, the internal stress conditions set up intestpiece blanks. I82 may be deduced by Lames solution for thick-walledcylinders (see Resistance of Materials, F. B. Seely, John Wiley andSons, New York, 1947 edition). When the internal pressure p1 exceedsexternal pressure po the radial stress s, at a radius r measured fromthe axis of symmetry will be compressive and the tangential stress s,will be tensile according to:

6 where a and b are the inner and outer radii of blanks 82. Since se ands, are principal stresses, the stress difference As at radius r'which isdeterminative of forced-hirefringence, is ygiven by:

Asf=ss6=2k/r2 (VI) where k is the constant rdeiined in'V above.

v Forced-birefringence B is `directly proportional to stress differenceAs and, therefore, may be equated as:

B=C/r2 (VII) where C is a constant equal to the product of k and amultiplier representing the optical strain sensitivity of the materialfof testpiece blanks 82. Assuming that thi-s pattern offorced-birefringence can be converted into a similar residual or biaspattern, the optimum -condition diagramed in FIG. 3 will result.

By means of furnace 80, the testpiece blanks l82 are brought evenly toan elevated temperature above the photoelastic critical temperature oftheir material, the pressure difference (pi-p0) is applied andequilibrium deformation of the blank is allowed to occur. Thereafter,the testpiece blanks 82 are cooled below the critical temperature beforereduction of the internal pressure. These process steps have been -foundto enforce the biasbirefringence pattern predicted by Equation VII uponeach of the testpiece blanks 82.

While the above formulae may be employed upon substitution of specificvalues for the constants, the direct optical presentation cf informationmay be taken advantage `o-f in engineering any given gauge model. Forexample, a sample testpiece blank was formed from s" sheet photoelasticplastic material to have inner and outer radii of Ms and SA,respectively. The sample was then viewed by :ordinary light transmittednormally through a polarizer and the lsample to a reflector and backthrough the :sample and an analyzer to the observer while the blank washeatedand subjected to a range of internal pressures. This proceduremade visible the formation and position `of the boundary fringe patternand enabled rapid standardization of the method step parameters.

It should be apparent that the improved omnidirectional photoelasticstrain gauge 'of this invention, as applied, is a very direct anduncomplicated exponential amplifier. A circle scribed on a workpiecesurface would be deformed as the workpiece was loaded and itsdeformation would be related to the workpiece surface strains, but suchydeformations would be too minute in practical cases to yield meaningfulinformation. 'Ihe de- :formation of the boundary fringe circle of the:gauge of this invention, however, is magnified by the optical strainsensitivity of the photoelastic material so that microscopic workpiecesurface strains lcan be read as millimeter displacements of the boundaryfringe position.

Gauge calibration is easily accomplished by applying a range of knownloading combinations to a dummy workpiece to which a sample gauge isattached.

While there Ihave been described what are at present considered to bethe preferred embodiments of this invention, it will be obvious to thoseskilled in the art that various changes and modifications may be madetherein without departing from the invention, and it is, therefore,aime-d in thev appended claims to cover all such changes andmodifications as fall within the true spirit and scope of the invention.

What is claimed is:

v 1. The method of producing a photoelastic testpiece having a radiallysymmetrical biasing pattern of bireifringence, which method comprisesthe steps of:

(a) forming from sheet photoelastic material a ilat circular testpieceblank having a concentric aperture,

7 (b) heating the blank to a temperature Within the photoelasticlcritical temperature range of the material, (c) subjecting the blank toan outwardly radially directed positive pressure applied at theperiphery of the aperture,

(d) cooling the blank below the photoelastic critical temperature rangewhile maintaining application of the radially directed pressure, and

(e) thereafter reducing the pressure whereby a substantially concentricresidual pattern of biasing birefringence is exhibited by the testpieceblank in the absence of external loading.

2. An improved strain sensitive element for onmidirectional photoelasticstrain -gauges for generating photoelastic patterns resolving vectorialsurface strains, which element comprises:

(a) a stratum [of photoelastic plastic material shaped to deiine a flatcircular testpiece with a concentric aperture, (b) said testpiece havinga radially symmetrical residual pattern of biasing birefringencepreformed therein exhibiting at yleast one circular boundary fringe whensaid testpiece is viewed by normal incidence polarized light in theabsence of external loading,

(c) said biasing birefringence at each point in said testpiece beinginversely proportional to the ydistance: of that point from the axis ofsymmetry of said testpiece, whereby total birefringence patternsproduced by said testpiece are polar coordinate plots of externallyimposed deformations of the outer periphery of said testpiece.

3. An improved omnidirecti-o-nal photoelastic strain gauge for thegeneration of photoelastic patterns resolving vectorial strains at thesurface of a workpiece, which gauge comprises:

(a) a stratum of photoelasti-c plastic material shaped to define a flatcircular testpiece with a concentric aperture,

and (b) attachment means attaching the outer peripheral edge of saidtestpiece to the workpiece surface, (c) said testpiece having a radiallysymmetrical residual pattern of bliasing vbirefringence preformedtherein exhibiting at least one circular boundary rfringe when saidtestpiece is viewed by normal incidence polarized light in the absenceof external loading, (d) said biasing birefringence at each point insaid testpiece being inversely proportional to the distance of thatpoint from the axis of symmetry of said testpiece, whereby the totalvbirefringence produced by said lgauge is a polar coordinate plot ofvectorial strains acting at the workpiece surface.

References lCited in the file of this patent UNITED STATES PATENTS2,332,674 Smith Oct. 26, 1943 3,034,341 Golubovic May l5, 1962

1. THE METHOD OF PRODUCING A PHOTOELASTIC TESTPIECE HAVING A RADIALLYSYMMETRICAL BIASING PATTERN OF BIREFRINGENCE, WHICH METHOD COMPRISES THESTEPS OF: (A) FORMING FROM SHEET PHOTOELASTIC MATERIAL A FLAT CIRCULARTESTPIECE BLANK HAVING A CONCENTRIC APERTURE, (B) HEATING THE BLANK TO ATEMPERATURE WITHIN THE PHOTOELASTIC CRITICAL TEMPERATURE RANGE OF THEMATERIAL, (C) SUBJECTING THE BLANK TO AN OUTWARDLY RADIALLY DIRECTEDPOSITIVE PRESSURE APPLIED AT THE PERIPHERY OF THE APERTURE, (D) COOLINGTHE BLANK BELOW THE PHOTOELASTIC CRITICAL TEMPERATURE RANGE WHILEMAINTAINING APPLICATION OF THE RADIALLY DIRECTED PRESSURE, AND (E)THEREAFTER REDUCING THE PRESSURE WHEREBY A SUBSTANTIALLY CONCENTRICRESIDUAL PATTERN OF BIASING BIREFRINGENCE IS EXHIBITED BY THE TESTPIECEBLANK IN THE ABSENCE OF EXTERNAL LOADING.